Yes, that's what "may" means: you may choose new targets for the copy, but if you don't, it keeps the original target.
One of the many rulings on the Gatherer page for Pyromancer's Goggles says as much:
The copy will have the same targets as the spell it’s copying unless you choose new ones. You may change any number of the targets, including all of them or none of them. If, for one of the targets, you can’t choose a new legal target, then it remains unchanged (even if the current target is illegal).